The (5, p)-arithmetic hyperbolic lattices in three dimensions : a dissertation in Mathematics, presented to the Massey University in partial fulfillment of the requirements for the degree of Doctor of Philosophy
| dc.contributor.advisor | Martin, Gaven | |
| dc.contributor.author | Salehi, Keyvan | |
| dc.date.accessioned | 2024-06-09T23:53:30Z | |
| dc.date.available | 2024-06-09T23:53:30Z | |
| dc.date.issued | 2024-02-10 | |
| dc.description.abstract | The group $Isom^+(\mathbb{H}^3)\cong PSL(2,\mathbb{C})$ contains an unlimited number of lattices of orientation-preserving isometries of hyperbolic 3-space (equivalently Kleinian groups of finite co-volume) that may be produced by using two elements of finite orders $p$ and $q$ as generators. For example, all but a finite number of $(p, 0)$-$(q, 0)$ orbifold Dehn surgery on any of the infinite number of hyperbolic two-bridge links (or knots if $p = q$) would have (orbifold) fundamental groups that are such uniform (co-compact) lattices. However, it was demonstrated in \cite{MM} that, up to conjugacy, two elements of finite order could generate only a finite number of arithmetic lattices. In fact, it is proved in \cite{MM} that there are only a finite number of {\em nearly arithmetic} groups, that is groups generated by two elements of finite order that are discrete subgroups of arithmetic groups and are not free on the two generators. The main result of this thesis is the determination of all the finitely many arithmetic lattices $\Gamma$ in the orientation preserving isometry group of hyperbolic $3$-space $\mathbb{H}^3$ generated by an element of order $5$ and an element of order $p\geq 2$, along with the determination of all the associated nearly arithmetic groups. These groups $\Gamma$ will have a presentation of the form \[ \Gamma\cong\langle f,g: f^5=g^p=w(f,g)=\cdots=1 \rangle \] In this Thesis, we find that necessarily \begin{itemize} \item $p\in \{2,3,4,5\}$ \item The total degree of the invariant trace field \[ k\Gamma=\mathbb{Q}(\{tr^2(h):h\in\Gamma\})\] is at most $6$ and at most $4$ for lattices. \item Each orbifold is either a two bridge link of slope $r/s$ surgered with $(5,0)$, $(p,0)$ orbifold Dehn surgery or a Heckoid group with rational slope $r/s\in [0,1]$ and $w(f,g)=(w_{r/s})^r$ with $r\in \{2,3,4,5\}$, and $w_{r/s}$ is a Farey word - described later. \end{itemize} For each such group, we find a discrete and faithful representation in $PSL(2,\mathbb{C})$, identify the rational slope $r/s$ and identify the associated number theoretic data. | |
| dc.identifier.uri | https://mro.massey.ac.nz/handle/10179/69754 | |
| dc.publisher | Massey University | en |
| dc.rights | The Author | en |
| dc.subject | Arithmetical algebraic geometry | en |
| dc.subject | Hyperbolic spaces | en |
| dc.subject | Lattice theory | en |
| dc.subject | Set theory | en |
| dc.subject.anzsrc | 490407 Mathematical logic, set theory, lattices and universal algebra | en |
| dc.title | The (5, p)-arithmetic hyperbolic lattices in three dimensions : a dissertation in Mathematics, presented to the Massey University in partial fulfillment of the requirements for the degree of Doctor of Philosophy | en |
| thesis.degree.discipline | phd | |
| thesis.degree.name | phd mathematics | |
| thesis.description.doctoral-citation-abridged | \bibitem{mz2} I. Agol, {\em Tameness of hyperbolic 3-manifolds}, arXiv:math/0405568, \bibitem{aft} L.V. Ahlfors, {\em Finitely generated Kleinian groups}, American Journal of Mathematics, {\bf 86}, (1964), 413--429. \bibitem{mz1} L.V. Ahlfors, {\em Fundamental polyhedrons and limit point sets of Kleinian groups}, Proceedings of the National Academy of Sciences of the United States of America, {\bf 55}, (1966), 251--254. \bibitem{ALSS} S. Aimi, D. Lee, S. Sakai, and M. Sakuma, {\em Classification of parabolic generating pairs of Kleinian groups with two parabolic generators}, preprint. \bibitem{AOPSY} H. Akiyoshi, K. Ohshika, J. Parker, M. Sakuma and H. Yoshida, {\em Classification of non-free Kleinain groups generated by two parabolic transformations}, preprint. \bibitem{ASWY} H. Akiyoshi, M. Sakuma, M. Wada, and Y. Yamashita, {\em Punctured torus groups and two bridge knot groups (I)}, Lecture Notes in Mathematics 1909, Springer-Verlag Berlin Heidelberg, 2007. \bibitem{Be} A. Beardon, {\em The geometry of discrete groups}, Springer--Verlag, 1983. \bibitem{orb} M. Boileau, B. Leeb, and J. Porti, {\em Geometrization of 3-dimensional orbifolds}, Annals of Math., {\bf 162}, (2005), 195--290. \bibitem{Bo} A. Borel {\em Commensurability classes and volumes of hyperbolic three-manifolds}, Ann. Sc. Norm. Pisa {\bf 8} (1981) 1 - 33. \bibitem{BZ} G. Burder and H. Zieschang, {\em Knots}, de Gruyter Studies in Math, {\bf 5}, Walter de Gruyter, 1985. \bibitem{mz3} D. Calegari, D. Gabai, {\em Shrinkwrapping and the taming of hyperbolic 3-manifolds}, J. American Math. Soc., {\bf 19}, (2006), 385--446, \bibitem{CDO} H. Cohen, F. Diaz Y Diaz, and M. Olivier, {\em Tables of octic fields with a quartic subfield}, preprint. \bibitem{Cooper} H. Cooper, {\em Discrete Groups and Computational Geometry}, Ph.D. Massey University, (2013), New Zealand. \bibitem{CM} MDE Conder, GJ Martin, {\em Cusps, triangle groups and hyperbolic $3$-folds} J. Austral. Math. Soc. Ser. A 55 (1993), 149--182. \bibitem{CMMO} M.D.E. Conder, C. Maclachlan, G.J. Martin, and E.A. O'Brien, {\em $2$--generator arithmetic Kleinian groups {\bf III}}. Math. Scand. 90 (2002), no. 2, 161--179. \bibitem{Di} F. Diaz Y Diaz, {\em Discriminant minimal et petis discriminants des corps de nombres de degre 7 avec cinq places reelles}, J. London Math. Soc. {\bf 38}, (1988), 33--46. \bibitem{DO} F. Diaz Y Diaz and M. Olivier, {\em Corps imprimitifs de degr\'e 9 de petit discriminant}, preprint. \bibitem{EMS1} A. Elzenaar, G.J. Martin and J. Schillewaert, {\em Approximations to the Riely slice}, arXiv:2111.03230, (2022). \bibitem{EMS2} A. Elzenaar, G.J. Martin and J. Schillewaert, {\em One complex dimensional moduli spaces: Keen - Series deformation theory}, Matrix, to appear, (2023). \bibitem{EMS3} A. Elzenaar, G.J. Martin and J. Schillewaert, {\em The combinatorics of Farey words and their traces}, arXiv:2204.08076 (2022). \bibitem{FR} Flammang and G. Rhin {\em Algebraic integers whose conjugates all lie in an ellipse}, Math. Comp. 74 (2005), no. 252, 2007-2015. \bibitem{GM1} F. W. Gehring and G. J. Martin, {\it Commutators, collars and the geometry of \mbox{M\"ob}ius groups}, J. d'Analyse Math., {\bf 63}, (1994), 175--219. \bibitem{GMMsemmat} F. W. Gehring, C. Maclachlan and G.J. Martin, {\em On the discreteness of the free product of finite cyclic groups}, Mitt. Math. Sem. Giessen, {\bf 228}, (1996), 9--15. \bibitem{GMM} F. W. Gehring, C. Maclachlan and G.J. Martin,{\em $2$--generator arithmetic Kleinian groups {\bf II}}, Bull. Lond. Math. Soc., {\bf 30}, (1998), 258--266. \bibitem{GMMR} F.W. Gehring, C. Maclachlan, G. Martin and A.W. Reid {\em Arithmeticity, discreteness, and volume}, Trans. Amer. Math. Soc., {\bf 349}, (1997), 3611--3643. \bibitem{HMR} M. Hagelberg, C. Maclachlan and G. Rosenberger {\em On discrete generalised triangle groups}, Proc. Edinburgh Math. Soc. {\bf 38}, (1995), 397 - 412. \bibitem{JR} J.W. Jones and D.P. Roberts, {\em A database of number fields}, LMS Journal of Computation and Mathematics, Cambridge University Press, 2014. \bibitem{Jor} T. J\o rgensen, {\em On discrete groups of Möbius transformations}, Amer. Journal of Math., {\bf 98}, (1976), 739--749. \bibitem{KS} L. Keen and C. Series, {\em The Riley slice of Schottky space}, Proc. London Math. Soc. 69 (1994), 72--90. \bibitem{KS2} L. Keen and C. Series, {\em Pleating coordinates for the maskit embedding of the Teichm\"uller space of punctured tori}, Topology, {\bf 32}, (1993), 719--749. \bibitem{KSM} L. Keen, B. Maskit and C. Series, {\em Geometric finiteness and uniqueness for Kleinian groups with circle packing limit sets}, J. Reine Angew. Math., {\bf 436}, (1993), 209--219. \bibitem{BW} W. B. R. Lickorish, {\em An Introduction to Knot Theory}, Springer--Verlag, 19897. \bibitem{MM} C. Maclachlan and G.J. Martin, {\em On 2--generator Arithmetic Kleinian groups.} J. Reine Angew. Math. {\bf 511}, (1999), 95--117 \bibitem{MM2} C. Maclachlan and G.J. Martin, {\em The non-compact generalised arithmetic triangle groups} Topology, {\bf 40}, (2001), 927--944. \bibitem{MM3} C. Maclachlan and G. J. Martin, {\em All Kleinian groups with two elliptic generators whose commutator is elliptic}, Math. Proc. Cambridge Philos. Soc. 135 (2003), no. 3, 413--420. \bibitem{MMpq} C. Maclachlan and G. J. Martin, {\em The $(p, q)$-arithmetic hyperbolic lattices; $p, q \geq 6$}, arXiv:1502.05453v1 \bibitem{MM6} C. Maclachlan and G. J. Martin, {\em The $(6,p)$-arithmetic hyperbolic lattices in dimension $3$}, Pure Appl. Math. Q., {\bf 7}, (2011), Special Issue: In honour of Frederick W. Gehring, Part 2, 365--382. \bibitem{MMM} C. Maclachlan, G.J. Martin, and J. McKenzie, {\em Arithmetic 2-generator Kleinian groups with quadratic invariant trace field.} New Zealand J. Math., {\bf 29}, (2000), 203--209. \bibitem{MR} C. Maclachlan and A. W. Reid, {\em The arithmetic of hyperbolic 3-manifolds}, Graduate Texts in Maths. Springer, 2003. \bibitem{MR1} C. Maclachlan and A.W. Reid {\em Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups}, Math. Proc. Camb. Phil. Soc., {\bf 102}, (1987), 251 -- 258. \bibitem{MR2} C. Maclachlan and A.W. Reid, {\em The arithmetic structure of tetrahedral groups of hyperbolic isometries.} Mathematika, {\bf 36}, (1989), 221--240. \bibitem{label1} G.J. Martin, {\em The Geometry and Arithmetic of Kleinian Groups}, arXiv, (2013). \bibitem{GKY} G. J. Martin and K. Salehi and Y. Yamashita {\em The $(4,p)$-arithmetic hyperbolic lattices, $p\geq 2$, in three dimensions}, arXive, (2022). \bibitem{Maskit} B. Maskit, {\em Kleinian groups}, Springer Verlag, 1989. \bibitem{MMarshall} T.H. Marshall and G.J. Martin, {\em Minimal co-volume hyperbolic lattices, II: Simple torsion in a Kleinian group}, Ann. of Math., {\bf 176}, (2012), 261--301. \bibitem{P1} J. Morgan and G. Tian, The geometrization conjecture. Clay Mathematics Monographs. Vol. 5. (2014) Cambridge, MA: Clay Mathematics Institute. \bibitem{Mull} H.P. Mullholland, {\em The product of $n$ complex homogeneous linear forms}, J. London Math. Soc. {\bf 35}. (1960), 241--250. \bibitem{NR} W.D. Neumann and A.W. Reid, {\em Arithmetic of hyperbolic manifolds.} Topology '90 (Columbus, OH, 1990), 273--310, Ohio State Univ. Math. Res. Inst. Publ., 1, de Gruyter, Berlin, 1992. \bibitem{Od} A.M. Odlyzko, {\em Some analytic estimates of class numbers and discriminants}, Invent Math. {\bf }, (1975). \bibitem{Rat} J.G. Ratcliffe, {\em Foundations of Hyperbolic Manifolds}, Springer--Verlag, 1994. \bibitem{P2} G. Perelman, {\em The entropy formula for the Ricci flow and its geometric applications}, (2002), arXiv: math/0211159. \bibitem{P3} G. Perelman, {\em Ricci flow with surgery on three-manifolds}. arXiv: math/0303109. \bibitem{P4} G. Perelman, {\em Finite extinction time for the solutions to the Ricci flow on certain three-manifolds}. arXiv: math/0307245. \bibitem{Rat} J.G. Ratcliffe, {\em Foundations of Hyperbolic Manifolds}, Springer--Verlag, (1994). \bibitem{RH} G. Rhin, { \em Approximants de Pad\'{e} et mesures effectives d’irrationalit\'{e}, S\'{e}minaire de Th\'{e}orie des Nombres}, Paris 1985-1986, Prog. Math. 71 (1987), 155-164. \bibitem{Rodgers} C.A. Rodgers, {\em The product of $n$ real homogeneous linear forms}, Acta Math. {\bf 82}, (1950),185--208. \bibitem{Rolfsen} Rolfsen, {\em Knots and links}, Corrected reprint of the 1976 original. Mathematics Lecture Series, 7. Publish or Perish, Inc., Houston, TX, (1990), arXiv439 pp. \bibitem {CJ} C. J. Smyth, { \em The mean value of totally real algebraic numbers}, Math. Comp. 42 (1984) 663-681. \bibitem{Tak} K. Takeuchi, {\it A characterization of arithmetic Fuchsian groups}, J. Math. Soc. Japan, {\bf 27}, (1975) 600-612. \bibitem{Take} K. Takeuchi, {\em Arithmetic triangle groups}, J. Math. Soc. Japan, {\bf 29}, (1977), 91--106. \bibitem{history} S. Thorgeirsson, {\em Hyperbolic geometry: history, models, and axioms}, https://uu.diva-portal.org/smash/get/diva2:729893/FULLTEXT01.pdf. \bibitem{Bill} W.P. Thurston, {\em Three-Dimensional Geometry and Topology}, Princeton University Press. \bibitem{S} I. Schur, {\em \"{U}ber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten.} Math. Zeit, {\bf 1}, (1918), 377 - 402. \bibitem{Stark} H.M. Stark, {\em Some effective cases of the Brauer-Siegel Theorem}, Invent. Math, {\bf 23}, (1974),135--152. \bibitem{Vig} M-F. Vigneras, {\em Arithm{\'e}tique des Alg{\`e}bres de Quaternions.} Lecture Notes in Mathematics, No. 800. Springer-Verlag, 1980. \bibitem{Z} Q. Zhang, {\em Two Elliptic Generator Kleinian Groups}, Ph.D. Thesis, Massey University, New Zealand, (2010). | |
| thesis.description.doctoral-citation-long | \bibitem{mz2} I. Agol, {\em Tameness of hyperbolic 3-manifolds}, arXiv:math/0405568, \bibitem{aft} L.V. Ahlfors, {\em Finitely generated Kleinian groups}, American Journal of Mathematics, {\bf 86}, (1964), 413--429. \bibitem{mz1} L.V. Ahlfors, {\em Fundamental polyhedrons and limit point sets of Kleinian groups}, Proceedings of the National Academy of Sciences of the United States of America, {\bf 55}, (1966), 251--254. \bibitem{ALSS} S. Aimi, D. Lee, S. Sakai, and M. Sakuma, {\em Classification of parabolic generating pairs of Kleinian groups with two parabolic generators}, preprint. \bibitem{AOPSY} H. Akiyoshi, K. Ohshika, J. Parker, M. Sakuma and H. Yoshida, {\em Classification of non-free Kleinain groups generated by two parabolic transformations}, preprint. \bibitem{ASWY} H. Akiyoshi, M. Sakuma, M. Wada, and Y. Yamashita, {\em Punctured torus groups and two bridge knot groups (I)}, Lecture Notes in Mathematics 1909, Springer-Verlag Berlin Heidelberg, 2007. \bibitem{Be} A. Beardon, {\em The geometry of discrete groups}, Springer--Verlag, 1983. \bibitem{orb} M. Boileau, B. Leeb, and J. Porti, {\em Geometrization of 3-dimensional orbifolds}, Annals of Math., {\bf 162}, (2005), 195--290. \bibitem{Bo} A. Borel {\em Commensurability classes and volumes of hyperbolic three-manifolds}, Ann. Sc. Norm. Pisa {\bf 8} (1981) 1 - 33. \bibitem{BZ} G. Burder and H. Zieschang, {\em Knots}, de Gruyter Studies in Math, {\bf 5}, Walter de Gruyter, 1985. \bibitem{mz3} D. Calegari, D. Gabai, {\em Shrinkwrapping and the taming of hyperbolic 3-manifolds}, J. American Math. Soc., {\bf 19}, (2006), 385--446, \bibitem{CDO} H. Cohen, F. Diaz Y Diaz, and M. Olivier, {\em Tables of octic fields with a quartic subfield}, preprint. \bibitem{Cooper} H. Cooper, {\em Discrete Groups and Computational Geometry}, Ph.D. Massey University, (2013), New Zealand. \bibitem{CM} MDE Conder, GJ Martin, {\em Cusps, triangle groups and hyperbolic $3$-folds} J. Austral. Math. Soc. Ser. A 55 (1993), 149--182. \bibitem{CMMO} M.D.E. Conder, C. Maclachlan, G.J. Martin, and E.A. O'Brien, {\em $2$--generator arithmetic Kleinian groups {\bf III}}. Math. Scand. 90 (2002), no. 2, 161--179. \bibitem{Di} F. Diaz Y Diaz, {\em Discriminant minimal et petis discriminants des corps de nombres de degre 7 avec cinq places reelles}, J. London Math. Soc. {\bf 38}, (1988), 33--46. \bibitem{DO} F. Diaz Y Diaz and M. Olivier, {\em Corps imprimitifs de degr\'e 9 de petit discriminant}, preprint. \bibitem{EMS1} A. Elzenaar, G.J. Martin and J. Schillewaert, {\em Approximations to the Riely slice}, arXiv:2111.03230, (2022). \bibitem{EMS2} A. Elzenaar, G.J. Martin and J. Schillewaert, {\em One complex dimensional moduli spaces: Keen - Series deformation theory}, Matrix, to appear, (2023). \bibitem{EMS3} A. Elzenaar, G.J. Martin and J. Schillewaert, {\em The combinatorics of Farey words and their traces}, arXiv:2204.08076 (2022). \bibitem{FR} Flammang and G. Rhin {\em Algebraic integers whose conjugates all lie in an ellipse}, Math. Comp. 74 (2005), no. 252, 2007-2015. \bibitem{GM1} F. W. Gehring and G. J. Martin, {\it Commutators, collars and the geometry of \mbox{M\"ob}ius groups}, J. d'Analyse Math., {\bf 63}, (1994), 175--219. \bibitem{GMMsemmat} F. W. Gehring, C. Maclachlan and G.J. Martin, {\em On the discreteness of the free product of finite cyclic groups}, Mitt. Math. Sem. Giessen, {\bf 228}, (1996), 9--15. \bibitem{GMM} F. W. Gehring, C. Maclachlan and G.J. Martin,{\em $2$--generator arithmetic Kleinian groups {\bf II}}, Bull. Lond. Math. Soc., {\bf 30}, (1998), 258--266. \bibitem{GMMR} F.W. Gehring, C. Maclachlan, G. Martin and A.W. Reid {\em Arithmeticity, discreteness, and volume}, Trans. Amer. Math. Soc., {\bf 349}, (1997), 3611--3643. \bibitem{HMR} M. Hagelberg, C. Maclachlan and G. Rosenberger {\em On discrete generalised triangle groups}, Proc. Edinburgh Math. Soc. {\bf 38}, (1995), 397 - 412. \bibitem{JR} J.W. Jones and D.P. Roberts, {\em A database of number fields}, LMS Journal of Computation and Mathematics, Cambridge University Press, 2014. \bibitem{Jor} T. J\o rgensen, {\em On discrete groups of Möbius transformations}, Amer. Journal of Math., {\bf 98}, (1976), 739--749. \bibitem{KS} L. Keen and C. Series, {\em The Riley slice of Schottky space}, Proc. London Math. Soc. 69 (1994), 72--90. \bibitem{KS2} L. Keen and C. Series, {\em Pleating coordinates for the maskit embedding of the Teichm\"uller space of punctured tori}, Topology, {\bf 32}, (1993), 719--749. \bibitem{KSM} L. Keen, B. Maskit and C. Series, {\em Geometric finiteness and uniqueness for Kleinian groups with circle packing limit sets}, J. Reine Angew. Math., {\bf 436}, (1993), 209--219. \bibitem{BW} W. B. R. Lickorish, {\em An Introduction to Knot Theory}, Springer--Verlag, 19897. \bibitem{MM} C. Maclachlan and G.J. Martin, {\em On 2--generator Arithmetic Kleinian groups.} J. Reine Angew. Math. {\bf 511}, (1999), 95--117 \bibitem{MM2} C. Maclachlan and G.J. Martin, {\em The non-compact generalised arithmetic triangle groups} Topology, {\bf 40}, (2001), 927--944. \bibitem{MM3} C. Maclachlan and G. J. Martin, {\em All Kleinian groups with two elliptic generators whose commutator is elliptic}, Math. Proc. Cambridge Philos. Soc. 135 (2003), no. 3, 413--420. \bibitem{MMpq} C. Maclachlan and G. J. Martin, {\em The $(p, q)$-arithmetic hyperbolic lattices; $p, q \geq 6$}, arXiv:1502.05453v1 \bibitem{MM6} C. Maclachlan and G. J. Martin, {\em The $(6,p)$-arithmetic hyperbolic lattices in dimension $3$}, Pure Appl. Math. Q., {\bf 7}, (2011), Special Issue: In honour of Frederick W. Gehring, Part 2, 365--382. \bibitem{MMM} C. Maclachlan, G.J. Martin, and J. McKenzie, {\em Arithmetic 2-generator Kleinian groups with quadratic invariant trace field.} New Zealand J. Math., {\bf 29}, (2000), 203--209. \bibitem{MR} C. Maclachlan and A. W. Reid, {\em The arithmetic of hyperbolic 3-manifolds}, Graduate Texts in Maths. Springer, 2003. \bibitem{MR1} C. Maclachlan and A.W. Reid {\em Commensurability classes of arithmetic Kleinian groups and their Fuchsian subgroups}, Math. Proc. Camb. Phil. Soc., {\bf 102}, (1987), 251 -- 258. \bibitem{MR2} C. Maclachlan and A.W. Reid, {\em The arithmetic structure of tetrahedral groups of hyperbolic isometries.} Mathematika, {\bf 36}, (1989), 221--240. \bibitem{label1} G.J. Martin, {\em The Geometry and Arithmetic of Kleinian Groups}, arXiv, (2013). \bibitem{GKY} G. J. Martin and K. Salehi and Y. Yamashita {\em The $(4,p)$-arithmetic hyperbolic lattices, $p\geq 2$, in three dimensions}, arXive, (2022). \bibitem{Maskit} B. Maskit, {\em Kleinian groups}, Springer Verlag, 1989. \bibitem{MMarshall} T.H. Marshall and G.J. Martin, {\em Minimal co-volume hyperbolic lattices, II: Simple torsion in a Kleinian group}, Ann. of Math., {\bf 176}, (2012), 261--301. \bibitem{P1} J. Morgan and G. Tian, The geometrization conjecture. Clay Mathematics Monographs. Vol. 5. (2014) Cambridge, MA: Clay Mathematics Institute. \bibitem{Mull} H.P. Mullholland, {\em The product of $n$ complex homogeneous linear forms}, J. London Math. Soc. {\bf 35}. (1960), 241--250. \bibitem{NR} W.D. Neumann and A.W. Reid, {\em Arithmetic of hyperbolic manifolds.} Topology '90 (Columbus, OH, 1990), 273--310, Ohio State Univ. Math. Res. Inst. Publ., 1, de Gruyter, Berlin, 1992. \bibitem{Od} A.M. Odlyzko, {\em Some analytic estimates of class numbers and discriminants}, Invent Math. {\bf }, (1975). \bibitem{Rat} J.G. Ratcliffe, {\em Foundations of Hyperbolic Manifolds}, Springer--Verlag, 1994. \bibitem{P2} G. Perelman, {\em The entropy formula for the Ricci flow and its geometric applications}, (2002), arXiv: math/0211159. \bibitem{P3} G. Perelman, {\em Ricci flow with surgery on three-manifolds}. arXiv: math/0303109. \bibitem{P4} G. Perelman, {\em Finite extinction time for the solutions to the Ricci flow on certain three-manifolds}. arXiv: math/0307245. \bibitem{Rat} J.G. Ratcliffe, {\em Foundations of Hyperbolic Manifolds}, Springer--Verlag, (1994). \bibitem{RH} G. Rhin, { \em Approximants de Pad\'{e} et mesures effectives d’irrationalit\'{e}, S\'{e}minaire de Th\'{e}orie des Nombres}, Paris 1985-1986, Prog. Math. 71 (1987), 155-164. \bibitem{Rodgers} C.A. Rodgers, {\em The product of $n$ real homogeneous linear forms}, Acta Math. {\bf 82}, (1950),185--208. \bibitem{Rolfsen} Rolfsen, {\em Knots and links}, Corrected reprint of the 1976 original. Mathematics Lecture Series, 7. Publish or Perish, Inc., Houston, TX, (1990), arXiv439 pp. \bibitem {CJ} C. J. Smyth, { \em The mean value of totally real algebraic numbers}, Math. Comp. 42 (1984) 663-681. \bibitem{Tak} K. Takeuchi, {\it A characterization of arithmetic Fuchsian groups}, J. Math. Soc. Japan, {\bf 27}, (1975) 600-612. \bibitem{Take} K. Takeuchi, {\em Arithmetic triangle groups}, J. Math. Soc. Japan, {\bf 29}, (1977), 91--106. \bibitem{history} S. Thorgeirsson, {\em Hyperbolic geometry: history, models, and axioms}, https://uu.diva-portal.org/smash/get/diva2:729893/FULLTEXT01.pdf. \bibitem{Bill} W.P. Thurston, {\em Three-Dimensional Geometry and Topology}, Princeton University Press. \bibitem{S} I. Schur, {\em \"{U}ber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten.} Math. Zeit, {\bf 1}, (1918), 377 - 402. \bibitem{Stark} H.M. Stark, {\em Some effective cases of the Brauer-Siegel Theorem}, Invent. Math, {\bf 23}, (1974),135--152. \bibitem{Vig} M-F. Vigneras, {\em Arithm{\'e}tique des Alg{\`e}bres de Quaternions.} Lecture Notes in Mathematics, No. 800. Springer-Verlag, 1980. \bibitem{Z} Q. Zhang, {\em Two Elliptic Generator Kleinian Groups}, Ph.D. Thesis, Massey University, New Zealand, (2010). | |
| thesis.description.name-pronounciation | keyvan salehi |
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